The nonlinear interaction of Tollmien–Schlichting waves and Taylor-Görtler vortices in curved channel flows

Abstract

It is known that a viscous fluid flow with curved streamlines can support both Tollmien-Schlichting and Taylor-Görtler instabilities. The question of which linear mode is dominant at finite values of the Reynolds numbers was discussed by Gibson & Cooke ( Q. Jl Mech. appl. Math . 27, 149 (1974)). In a situation where both modes are possible on the basis of linear theory a nonlinear theory must be used, however, to determine the effect of the interaction of the instabilities. The details of this interaction are of practical importance because of its possible catastrophic effects on mechanisms used for laminar flow control. Here this interaction is studied in the context of fully developed flows in curved channels. Apart from technical differences associated with boundary-layer growth the structures of the instabilities in this flow can be very similar to those in the practically more important external boundary-layer situation. The interaction is shown to have two distinct phases depending on the size of the input disturbances. At very low amplitudes two oblique Tollmien–Schlichting waves interact with a Görtler vortex in such a manner that the scaled amplitudes become infinite at a finite time. This type of interaction is described by ordinary differential amplitude equations with quadratic nonlinearities. A stronger type of interaction occurs at larger input disturbance amplitudes and leads to a more complicated type of evolution equation. The solution of these equations now depends critically on the orientation of the wavefronts of the Tollmien–Schlichting waves to the Görtler vortex. Thus, if the angle between the directions of the vortex and the waves is greater than 41.6° this stronger interaction again terminates in a singularity at a finite time; otherwise the breakdown is exponential, taking an infinite time. Moreover, the stronger interaction can take place in the absence of curvature, in which case the longitudinal vortex is entirely driven by the Tollmien–Schlichting waves.